quadm (p42, negative)

Average Error: 1.6 → 1.6

Time: 32.8s

Precision: 64

Math

\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]

↓

\[\left(\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(\left(4\right) \cdot a\right), c\right)\right)\right)}\right)\right)}{\left(1.0\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(\left(2\right) \cdot a\right)}\right)\]

C

double f(double a, double b, double c) {
        double r1259757 = b;
        double r1259758 = -r1259757;
        double r1259759 = r1259757 * r1259757;
        double r1259760 = 4.0;
        double r1259761 = /* ERROR: no posit support in C */;
        double r1259762 = a;
        double r1259763 = c;
        double r1259764 = r1259762 * r1259763;
        double r1259765 = r1259761 * r1259764;
        double r1259766 = r1259759 - r1259765;
        double r1259767 = sqrt(r1259766);
        double r1259768 = r1259758 - r1259767;
        double r1259769 = 2.0;
        double r1259770 = /* ERROR: no posit support in C */;
        double r1259771 = r1259770 * r1259762;
        double r1259772 = r1259768 / r1259771;
        return r1259772;
}

↓

double f(double a, double b, double c) {
        double r1259773 = b;
        double r1259774 = -r1259773;
        double r1259775 = r1259773 * r1259773;
        double r1259776 = /*Error: no posit support in C */;
        double r1259777 = 4.0;
        double r1259778 = /* ERROR: no posit support in C */;
        double r1259779 = a;
        double r1259780 = r1259778 * r1259779;
        double r1259781 = c;
        double r1259782 = /*Error: no posit support in C */;
        double r1259783 = /*Error: no posit support in C */;
        double r1259784 = sqrt(r1259783);
        double r1259785 = r1259774 - r1259784;
        double r1259786 = 1.0;
        double r1259787 = /* ERROR: no posit support in C */;
        double r1259788 = r1259785 / r1259787;
        double r1259789 = 2.0;
        double r1259790 = /* ERROR: no posit support in C */;
        double r1259791 = r1259790 * r1259779;
        double r1259792 = r1259787 / r1259791;
        double r1259793 = r1259788 * r1259792;
        return r1259793;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 1.6

   \[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
2. Using strategy rm

3. Applied -p16-rgt-identity-expand 1.6

   \[\leadsto \frac{\left(\color{blue}{\left(\left(-b\right) - \left(0.0\right)\right)} - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]

4. Applied associate--l- 1.6

   \[\leadsto \frac{\color{blue}{\left(\left(-b\right) - \left(\frac{\left(0.0\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)\right)}}{\left(\left(2\right) \cdot a\right)}\]

5. Simplified 1.6

   \[\leadsto \frac{\left(\left(-b\right) - \color{blue}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(\left(4\right) \cdot a\right) \cdot c\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
6. Using strategy rm

7. Applied introduce-quire 1.6

   \[\leadsto \frac{\left(\left(-b\right) - \left(\sqrt{\left(\color{blue}{\left(\left(\left(b \cdot b\right)\right)\right)} - \left(\left(\left(4\right) \cdot a\right) \cdot c\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]

8. Applied insert-quire-fdp-sub 1.6

   \[\leadsto \frac{\left(\left(-b\right) - \left(\sqrt{\color{blue}{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(\left(4\right) \cdot a\right), c\right)\right)\right)}}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
9. Using strategy rm

10. Applied *p16-rgt-identity-expand 1.6

    \[\leadsto \frac{\color{blue}{\left(\left(\left(-b\right) - \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(\left(4\right) \cdot a\right), c\right)\right)\right)}\right)\right) \cdot \left(1.0\right)\right)}}{\left(\left(2\right) \cdot a\right)}\]

11. Applied p16-times-frac 1.6

    \[\leadsto \color{blue}{\left(\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(\left(4\right) \cdot a\right), c\right)\right)\right)}\right)\right)}{\left(2\right)}\right) \cdot \left(\frac{\left(1.0\right)}{a}\right)}\]
12. Using strategy rm

13. Applied p16-*-un-lft-identity 1.6

    \[\leadsto \left(\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(\left(4\right) \cdot a\right), c\right)\right)\right)}\right)\right)}{\color{blue}{\left(\left(1.0\right) \cdot \left(2\right)\right)}}\right) \cdot \left(\frac{\left(1.0\right)}{a}\right)\]

14. Applied *p16-rgt-identity-expand 1.6

    \[\leadsto \left(\frac{\color{blue}{\left(\left(\left(-b\right) - \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(\left(4\right) \cdot a\right), c\right)\right)\right)}\right)\right) \cdot \left(1.0\right)\right)}}{\left(\left(1.0\right) \cdot \left(2\right)\right)}\right) \cdot \left(\frac{\left(1.0\right)}{a}\right)\]

15. Applied p16-times-frac 1.6

    \[\leadsto \color{blue}{\left(\left(\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(\left(4\right) \cdot a\right), c\right)\right)\right)}\right)\right)}{\left(1.0\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(2\right)}\right)\right)} \cdot \left(\frac{\left(1.0\right)}{a}\right)\]

16. Applied associate-*l* 1.6

    \[\leadsto \color{blue}{\left(\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(\left(4\right) \cdot a\right), c\right)\right)\right)}\right)\right)}{\left(1.0\right)}\right) \cdot \left(\left(\frac{\left(1.0\right)}{\left(2\right)}\right) \cdot \left(\frac{\left(1.0\right)}{a}\right)\right)}\]

17. Simplified 1.6

    \[\leadsto \left(\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(\left(4\right) \cdot a\right), c\right)\right)\right)}\right)\right)}{\left(1.0\right)}\right) \cdot \color{blue}{\left(\frac{\left(1.0\right)}{\left(\left(2\right) \cdot a\right)}\right)}\]

18. Final simplification 1.6

    \[\leadsto \left(\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(\left(4\right) \cdot a\right), c\right)\right)\right)}\right)\right)}{\left(1.0\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(\left(2\right) \cdot a\right)}\right)\]

Reproduce

herbie shell --seed 2019164 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  (/.p16 (-.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))